set. That of Part I. is an introduction to the employment of imaginary quantities in trigonometric developments, while that of Part II. is an introduction to the higher forms of solid geometry. VI. To the usual list of subjects treated, has been added 1 chapter on the theory of polygons. This theory is closely con nected with a variety of subjects, including geometry, quaternions, mechanics, graphical statics, surveying, and navigation, and there- CHAPTER II. OF THE TRIGONOMETRIC FUNCTIONS.. The sine, tangent, and secant, 11. Functions of unlimited angles, 15. The cosine, cotangent, and cosecant, 19. Values for the second, third, and fourth quadrants, 22. Special values of the trigonometric func- tions, 26. Angles corresponding to given functions, 27. Relations CHAPTER IV. RELATIONS BETWEEN FUNCTIONS OF SEVERAL ANGLES.. 44 one side, 62. Case II. Given two sides and the angle opposite one of them, 64. Case III. Given the three sides, 66. Case IV. Given two Co-ordinates of a point, 75. Definition of direction, 77. Projections of lines, 78. Algebraic addition of lines, 80. Theorems of projections of the sides of a polygon, 81. Areas of polygons, 86. CHAPTER VIII. TRIGONOMETRIC DEVELOPMENTS.... Development of sines and cosines in powers of the angle, 92. Imagi nary exponentials, 96. Sines and cosines of multiple arcs, 97. Powers of the cosine, 99. Powers of the sine, 101. Trigonometric forms of Preliminary notions, 113. Fundamental equations, 116. Polar tri- Fundamental definitions and theorems, 129. triangles, 130, Napier's rules, 132. Relations among four parts, 133. CHAPTER III. TRANSFORMATION OF THE FORMULA OF SPHERICAL TRIGO- Expressions when the three sides or the three angles are given, 140. Cagnoli's equation, 143. Gauss's equations, 144. Napier's analogies, CHAPTER IV. MISCELLANEOUS APPLICATIONS Geographical applications, 148. Geometrical applications, 150. Rec- tangular lines and planes, 150. Direction of a line in space, 154. Relation to latitude and longitude, 156. Position of a point, 156. Polar distance and longitude, 157. Rectangular co-ordinates, 157. Projection of one line upon another, 161. Plane triangles in space, ELEMENTS OF TRIGONOMETRY PART I. PLANE TRIGONOMETRY. CHAPTER I. OF GONIOMETRY, OR THE MEASURE OF ANGLES. 1. Definition. Trigonometry is that branch of geometry in which the relations of lines and angles are treated by algebraic methods. 2. Def. An angle is the figure formed by two straight lines emanating from the same point, called the vertex of the angle. Def. The lines which form an angle are called its sides. 3. Measures of Angles. An angle is measured by the length of a circular arc having its centre at the vertex of the angle and its ends on the sides of the angle. If the angle to be measured is AOB, we conceive that with an arbitrary radius Oa an arc is drawn from a to b. We regard as the positive direction that in which the arc is described by a motion 1 A opposite to that of the hands of a watch, and as the negative direction that in which the hands move. Hence we may consider the angle as measured either by the arc ab considered as positive, or the conjugate arc aMb considered as negative. The numerical sum of these two arcs is equal to a circumference. As an example of the use of algebraic signs, we may mention. their application to the latitude of places to distinguish them as north and south. Thus, a city in 42° north latitude is said to have a latitude of 42°, and one 42° south of the equator is said to be in latitude - 42°. The absolute length of the arc will depend not only upon the magnitude of the angle, but upon the radius with which the arc is drawn. To avoid ambiguity from this cause, the unit of arc is supposed to be some fixed fraction of the circumference, and therefore greater the greater the radius. The arc is then indicated by the number of units and parts of a unit which it contains, and this number is the same for the same angle whatever the radius may be. To indicate the angle corresponding to any arc we call it the angle of the arc, or, for brevity, the angle-arc. 4. The Sexagesimal Division. The following is the usual division: The circumference is divided into 360 units, called degrees; Each degree is divided into 60 minutes; Each minute is divided into 60 seconds. Then 1 circumference 360° = 21 600′ = 1 296 000"; = 90°. This is called the sexagesimal division of the circle. 5. The Centesimal Division. The sexagesimal division of the circle is by no means so convenient as one in which each unit is 10 times or 100 times greater than the next smaller unit. The centesimal division was introduced by the French geometers at the time of the Revolution. In this system The circumference is divided into 400 grades; The grade is divided into 100 minutes ; The minute is divided into 100 seconds. |